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To heat water using a solar heater, the first step is to calculate the heat energy required. This is done using the formula: \[ Q = mc\Delta T \]Where:

• \( Q \) is the amount of heat energy
• \( m \) is the mass of the water
• \( c \) is the specific heat capacity of water
• \( \Delta T \) is the change in temperature

To find the mass \( m \), we convert the volume of the water into mass using the fact that 1 liter of water equals 1 kilogram. For 300 liters of water, we get: \[ m = 300 \text{ kg} \]Next, the change in temperature \( \Delta T \) is the difference between the final and initial temperatures: \[ \Delta T = 50 \degree C - 15 \degree C = 35 \degree C \]Now, using the specific heat capacity of water as \( 4.18 \times 10^3 \text{ J/kg}^\circ \text{C} \), we can calculate the total heat energy required: \[ Q = 300 \text{ kg} \times 4.18 \times 10^3 \text{ J/kg}^\circ \text{C} \times 35 \degree C = 43,890,000 \text{ J} \]

Solar radiation is the key source of energy for the solar heater. It is measured in watts per square meter (\( \text{W/m}^2 \)). For this exercise, the solar radiation received is \( 240 \text{ W/m}^2 \).To find the total solar energy input over the given heating period (12 hours), we convert the time into seconds since power (watts) is energy per unit time in seconds:\[ 12 \text{ hours} \times 3600 \text{ seconds/hour} = 43,200 \text{ seconds} \]Then, the total solar energy falling per square meter is calculated by multiplying the solar radiation by the total time in seconds:\[ E_{solar} = 240 \text{ W/m}^2 \times 43,200 \text{ seconds} = 10,368,000 \text{ J/m}^2 \]This energy represents the maximum potential solar energy available. However, not all of it is effectively used due to efficiency losses, which we will address next.

Specific heat capacity is a property of a material that tells how much energy is needed to raise the temperature of 1 kilogram of the material by 1 degree Celsius.For water, the specific heat capacity \( c \) is \( 4.18 \times 10^3 \text{ J/kg}^\circ \text{C} \). This high value means water can store a lot of heat energy, which is why it is a preferred substance in thermal applications such as solar heaters. It ensures consistent heating over a longer period since a substantial amount of energy is absorbed by the water before it changes temperature significantly.When you use this value in our heat energy calculation (\[ Q = mc\Delta T \]), it helps determine the total energy in joules required to achieve the desired temperature change.

Energy efficiency is crucial in determining how effectively the solar heater converts solar energy into usable heat. In this case, the solar collector has an efficiency of 65%. This means only 65% of the solar radiation energy is converted into heat energy for the water.To find the effective energy collected, multiply the total solar energy by the efficiency percentage:\[ E_{collected} = 10,368,000 \text{ J/m}^2 \times 0.65 = 6,739,200 \text{ J/m}^2 \]This helps in identifying how much useful energy is available to heat the water. Finally, we determine the required area of the solar panel to meet the necessary heat energy using the formula:\[ A = \frac{Q}{E_{collected}} \approx 6.51 \text{ m}^2 \]This ensures that we have enough capturing area to meet the heating demands, considering the efficiency losses.